Related papers: Large bias for integers with prime factors in arit…

Given a zero-free region and an averaged zero-density estimate over all Dirichlet $L$-functions modulo $q\in\mathbb{N}$, we refine the error terms of the prime number theorem in all and almost all short arithmetic progressions. For example,…

A permutation of the positive integers avoiding monotone arithmetic progressions of length $4$ with odd common difference was constructed in (LeSaulnier and Vijay, 2011). We generalise this result and show that for each $k\geq 1$, there…

We prove specific biases in the number of occurrences of parts belonging to two different residue classes $a$ and $b$, modulo a fixed non-negative integer $m$, for the sets of unrestricted partitions, partitions into distinct parts, and…

We show that for all large enough $x$ the interval $[x,x+x^{1/2}\log^{1.39}x]$ contains numbers with a prime factor $p > x^{18/19}.$ Our work builds on the previous works of Heath-Brown and Jia (1998) and Jia and Liu (2000) concerning the…

If $a$ and $b$ are integers with $b>a>1$, we completely characterize ``long'' arithmetic progressions in the sumsets of the geometric progressions $1, a, a^2, a^3, \ldots$ and $1, b, b^2, b^3, \ldots$. Our proofs utilize recent applications…

Let $q\ge 3$ be a non-exceptional modulus $q\ge3$, and let $a$ be a positive integer coprime with $q$. For any $\epsilon>0$, there exists $\alpha>0$ (computable), such that for all $x\ge \alpha (\log q)^2$, the interval $\left[…

We show that for any relatively prime integers $1\leq p<q$ and for any finite $A \subset \mathbb{Z}$ one has $$|p \cdot A + q \cdot A | \geq (p + q) |A| - (pq)^{(p+q-3)(p+q) + 1}.$$

We prove asymptotic formulas for the complex coefficients of $(\zeta q;q)_\infty^{-1}$, where $\zeta$ is a root of unity, and apply our results to determine secondary terms in the asymptotics for $p(a,b,n)$, the number of integer partitions…

Let $[\, \cdot\,]$ be the floor function. In this paper we show that every sufficiently large positive integer $N$ can be represented in the form \begin{equation*} N=[p_1\log p_1]+[p_2\log p_2]+[p_3\log p_3], \end{equation*} where $p_1,\,…

We study the distribution of partition parts in arithmetic progressions and find asymptotic results that capture all exponentially growing terms. This is accomplished by studying the behavior of non-modular Eisenstein series that appear in…

Let $\{p_j(n)\}_{j=1}^{\omega(n)}$ denote the increasing sequence of distinct prime factors of an integer $n$. We provide details for the proof of a statement of Erd\H{o}s implying that, for any function $\xi(n)$ tending to infinity with…

Let $p_n$ denote the $n$-th prime. For any $m\geq 1$, there exist infinitely many $n$ such that $p_{n}-p_{n-m}\leq C_m$ for some large constant $C_m>0$, and $$p_{n+1}-p_n\geq \frac{c_m\log n\log\log n\log\log\log\log n}{\log\log\log n}, $$…

In recent work, we considered the frequencies of patterns of consecutive primes $\pmod{q}$ and numerically found biases toward certain patterns and against others. We made a conjecture explaining these biases, the dominant factor in which…

By Maynard's theorem and the subsequent improvements by the Polymath Project, there exists a positive integer $b\leq 246$ such that there are infinitely many primes $p$ such that $p+b$ is also prime. Let $P_1,...,P_t\in \mathbb{Z}[y]$ with…

Chebyshev was the first to observe a bias in the distribution of primes in residue classes. The general phenomenon is that if $a$ is a nonsquare\mod q and $b$ is a square\mod q, then there tend to be more primes congruent to $a\mod q$ than…

We obtain an analog of the Montgomery-Hooley asymptotic formula for the variance of the number of primes in arithmetic progressions. In the present paper the moduli are restricted to the sequences of integer parts $[F(n)]$, where $F(t) =…

For $k\geq 1$ an integer and $x\geq 1$ a real number, let $\pi_k(x)$ be the number of integers smaller than $x$ having exactly $k$ distinct prime divisors. Building on recent work of Matom\"aki and Radziwi\l\l, we investigate the asymptotic…

We determine the asymptotic density $\delta_k$ of the set of ordered $k$-tuples $(n_1,...,n_k)\in \N^k, k\ge 2$, such that there exists no prime power $p^a$, $a\ge 1$, appearing in the canonical factorization of each $n_i$, $1\le i\le k$,…

Let $a_0=b_0=0$ and $0<a_1\leq b_1<a_2\leq b_2<\ldots\leq b_{n}$ be integers. Let $Q\left(x;\bigcup_{j=1}^{n}[a_j,b_j]\right)$ be the number of integers between $1$ and $x$ such that all exponents in their prime factorization are in…

We introduce a new arithmetic function $a(n)$ defined to be the number of random multiplications by residues modulo $n$ before the running product is congruent to 0 modulo $n$. We give several formulas for computing the values of this…